A primer on mapping class groups by Farb B., Margalit D.

The learn of the mapping type team Mod(S) is a classical subject that's experiencing a renaissance. It lies on the juncture of geometry, topology, and crew concept. This publication explains as many vital theorems, examples, and strategies as attainable, quick and at once, whereas while giving complete information and preserving the textual content approximately self-contained. The e-book is acceptable for graduate students.A Primer on Mapping classification teams starts by way of explaining the most group-theoretical homes of Mod(S), from finite iteration via Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. alongside the way in which, valuable gadgets and instruments are brought, resembling the Birman specific series, the advanced of curves, the braid workforce, the symplectic illustration, and the Torelli staff. The ebook then introduces Teichmller area and its geometry, and makes use of the motion of Mod(S) on it to turn out the Nielsen-Thurston type of floor homeomorphisms. issues contain the topology of the moduli house of Riemann surfaces, the relationship with floor bundles, pseudo-Anosov concept, and Thurston's method of the type.

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The starting point is the case of nonseparating simple closed curves, and the inductive step is Example 5: cutting along the first few arcs, the next arc becomes a nonseparating arc on the cut surface. Note that Example 1 is the case k = 2. One can also prove by induction that every chain in Sg of even length is nonseparating, and so such chains must be topologically equivalent. We remark that the homeomorphism representing the change of coordinates in each of the six examples above can be taken to be orientation preserving.

The way we will compute Mod(S0,3 ) is to understand its action on some fixed arc in S0,3 . The surface obtained by cutting S0,3 along this arc is a punctured disk, and so we will be able to apply the Alexander lemma. This is in general how we use the cutting procedure for surfaces in order to perform inductive arguments. In this section it will be convenient to think of S0,3 as a sphere with three marked points (instead of three punctures). In order to determine Mod(S0,3 ) we first need to understand simple proper arcs in S0,3 .

44 CHAPTER 1 In fact a stronger, relative result holds: if two homeomorphisms are homotopic relative to ∂S then they are isotopic relative to ∂S. 8. 12 also holds when S has finitely many marked points. In that case, we need to expand our list of counterexamples to include a sphere with one or two marked points. 2 H OMEOMORPHISMS VERSUS DIFFEOMORPHISMS It is sometimes convenient to work with homeomorphisms and sometimes convenient to work with diffeomorphisms. For example it is easier to construct the former but we can apply differential topology to the latter.